# Stochastic gradient descent in 'stronger' settings

I am minimzing a function $$F(x) = \mathbb E(f(x,\Xi))$$ where $$\Xi$$ is some random value, by a stochastic gradient descent that generates a random number $$\xi$$ from the distribution of $$\Xi$$ at each iterate and uses the gradient $$g_\xi(x) = \nabla_f(x,\xi).$$

This works well, and there are a lot of theory that essentialy says that if $$g(x) = \mathbb E(g_\Xi(x))$$ is bouded, if $$F$$ is strongly convex and smooth, etc. we have some convergence results.

The classical framework deals with the case where $$f(.,\xi)$$ are all convex functions but with different minimums in $$x$$. In my special case, all $$f(.,\xi)$$ are convex, and have minimums that are not a point, but rather a variety (x is multivariate), and the global minimum of $$F$$, which is a point, lies at the intersection of the minimas of $$f(.,\xi)$$'s.

This is a huge plus for the convergence of the stochastic gradient descent, however I am not sure how to deal with it. Was it treated somewhere in the literature ?